Regularized area-level modelling for robust small area estimation in the presence of unknown covariate measurement errors

An approach to model-based small area estimation under covariate measurement errors is presented. Using a min-max approach, we proof that regularized regression coefficient estimation is equivalent to robust optimization under additive noise. Applying this equivalence, the Fay-Herriot model is extended by l1-norm, squared l2-norm and elastic net regularizations as robustification against design matrix perturbations. This allows for reliable area-statistic estimates without distributive information about the measurement errors. A best predictor and a Jackknife estimator of the mean squared error are presented. The methodology is evaluated in a simulation study under multiple measurement error scenarios to support the theoretical findings. A comparison to other robust small area approaches is conducted. An empirical application to poverty mapping in the US is provided. Estimated economic figures from the US Census Bureau and crime records from the Uniform Crime Reporting Program are used to model the number of citizens below the federal poverty threshold.